Question

question
express in simplest radical form.
\sqrt{125}

Explanation:

Step1: Factor 125 into prime factors

We know that \(125 = 5\times5\times5=5^{3}\), so \(\sqrt{125}=\sqrt{5^{3}}\).

Step2: Use the property of square roots \(\sqrt{a^{m}}=a^{\frac{m}{2}}\) (for \(a\geq0\))

\(\sqrt{5^{3}} = 5^{\frac{3}{2}}=5^{1 + \frac{1}{2}}\).
According to the exponent rule \(a^{m + n}=a^{m}\times a^{n}\), we have \(5^{1+\frac{1}{2}}=5^{1}\times5^{\frac{1}{2}}\).
Since \(5^{\frac{1}{2}}=\sqrt{5}\), then \(5\times\sqrt{5} = 5\sqrt{5}\).
Or we can also use the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)) directly:
\(\sqrt{125}=\sqrt{25\times5}\), and since \(\sqrt{25\times5}=\sqrt{25}\times\sqrt{5}\), and \(\sqrt{25} = 5\), so \(\sqrt{125}=5\sqrt{5}\).

Answer:

\(5\sqrt{5}\)